where \(\bar{z} = \mathbb {E}[\mbox{$z$}]\) ; see [1]}. The constraint asks for low covariance between \(\mbox{$z$}\)  and the classifier. For \(n = 1\) , \(t = 0.25\) , and a distribution \(P\)  of \((\mbox{$x$}, \mbox{$y$}, \mbox{$z$})\)  assigning probability \(1/2\)  to \((-1, -1, 0)\)  and probability \(1/2\)  to \((1,1,1)\) , we obtain the set of minimizers \(\lbrace 1/2\rbrace  \times [-1/2, 1/2]\) . However, if the distribution is slightly altered to assigning probability \(1/2\)  to \((-1, -1, 0)\) , probability \(1/2 - 1/\nu \)  to \((1,1,1)\) , and probability \(1/\nu \)  to \((\nu , 1, 1)\) , then the corresponding minimizers are unique for sufficiently large \(\nu \)  and they converge to the incorrect \((1/4, -3/4)\) . This takes place even though the approximating probability distribution converges weakly to the actual distribution \(P\) .

